Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors

Abstract

Let \(\Kl(1;q)\) denote the normalized Kloosterman sum modulo \(q\). We prove that, for each sign, there are \(\gg X/\log X\) square-free moduli \(q\in(X,2X]\) with at most four prime factors for which \(\Kl(1;q)\) has that sign. The proof combines the Selberg sieve with equidistribution and mean-square estimates for Kloosterman sums; the uniform estimate for shifted sieve weights established in Part~I controls the contribution of moduli having small prime factors.